{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T07:35:56Z","timestamp":1774596956381,"version":"3.50.1"},"reference-count":42,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2020,10,20]],"date-time":"2020-10-20T00:00:00Z","timestamp":1603152000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The questions of the one-value solvability of an inverse boundary value problem for a mixed type integro-differential equation with Caputo operators of different fractional orders and spectral parameters are considered. The mixed type integro-differential equation with respect to the main unknown function is an inhomogeneous partial integro-differential equation of fractional order in both positive and negative parts of the multidimensional rectangular domain under consideration. This mixed type of equation, with respect to redefinition functions, is a nonlinear Fredholm type integral equation. The fractional Caputo operators\u2019 orders are smaller in the positive part of the domain than the orders of Caputo operators in the negative part of the domain under consideration. Using the method of Fourier series, two systems of countable systems of ordinary fractional integro-differential equations with degenerate kernels and different orders of integro-differentation are obtained. Furthermore, a method of degenerate kernels is used. In order to determine arbitrary integration constants, a linear system of functional algebraic equations is obtained. From the solvability condition of this system are calculated the regular and irregular values of the spectral parameters. The solution of the inverse problem under consideration is obtained in the form of Fourier series. The unique solvability of the problem for regular values of spectral parameters is proved. During the proof of the convergence of the Fourier series, certain properties of the Mittag\u2013Leffler function of two variables, the Cauchy\u2013Schwarz inequality and Bessel inequality, are used. We also studied the continuous dependence of the solution of the problem on small parameters for regular values of spectral parameters. The existence and uniqueness of redefined functions have been justified by solving the systems of two countable systems of nonlinear integral equations. The results are formulated as a theorem.<\/jats:p>","DOI":"10.3390\/axioms9040121","type":"journal-article","created":{"date-parts":[[2020,10,20]],"date-time":"2020-10-20T09:28:23Z","timestamp":1603186103000},"page":"121","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":35,"title":["Inverse Problem for a Mixed Type Integro-Differential Equation with Fractional Order Caputo Operators and Spectral Parameters"],"prefix":"10.3390","volume":"9","author":[{"given":"Tursun K.","family":"Yuldashev","sequence":"first","affiliation":[{"name":"Uzbek-Israel Joint Faculty of High Technology and Engineering Mathematics, National University of Uzbekistan, Tashkent 100174, Uzbekistan"}]},{"given":"Erkinjon T.","family":"Karimov","sequence":"additional","affiliation":[{"name":"V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, Tashkent 100174, Uzbekistan"}]}],"member":"1968","published-online":{"date-parts":[[2020,10,20]]},"reference":[{"key":"ref_1","unstructured":"Samko, S.G., Kilbas, A.A., and Marichev, O.I. (1993). Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Carpinteri, A., and Mainardi, F. (1997). Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics. Fractals and Fractional Calculus in Continuum Mechanics, Springer.","DOI":"10.1007\/978-3-7091-2664-6"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Area, I., Batarfi, H., Losada, J., Nieto, J.J., Shammakh, W., and Torres, A. (2015). On a fractional order Ebola epidemic model. Adv. Differ. Equ., 278.","DOI":"10.1186\/s13662-015-0613-5"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Hussain, A., Baleanu, D., and Adeel, M. (2020). 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